Historical ways of *presenting* or axiomatizing the notion of a topological space

Question

I've heard rumors in a couple of places that the modern presentation of a topology as $(X, \tau)$ where $X$ is our topological space and $\tau$ is our set of open sets is "somewhat" new and a simplification of how topologies used to be presented. We can also pick simple variants of this such as $(X, B)$ where $B$ is a basis or subbase and identify them with the modern presentation.

What are the different ways that topologies have been presented or defined over time?

This youtube video is a review of an old book called Elements of Point Set Topology, which defines a system of neighborhoods at each point. The neighborhood presentation is sort of confusing, but I think the way to map it back to a more modern presentation is to note that the union of all neighborhoods for every point forms a basis, based on this article on ProofWiki, but I am a little confused as to the intuition behind a neighborhood system.

My question is similar to this one, but is specifically about the old definitions that were out-competed in a sense.

Answer 1

An early definition of topology is given in the book Kuratowski, Topology I (first edition, 1933). It is defined in terms of a "closure operator", $X\mapsto\overline{X}$ acting on the set of all subsets of a set, which satisfies 3 axioms: $$\overline{(X\cup Y)}=\overline{X}\cup\overline{Y},\quad \overline{\overline{X}}=\overline{X},$$ and $\overline{X}=X$ when $X$ is one point or empty.

The modern definition was popularised by Bourbaki, and in English by Kelley, in his book General topology, in the early 1950s, but it was already in use the 1920s, in the work of P. Aleksandrov and Urysohn, for example.

Aleksandrov, in his comments to the Russian edition of his work with Urysohn, credits the modern axioms to Hausdorff, Grundzuge der Mengenlehre, 1914, though his set of axioms included the "Hausdorff axiom" which nowadays is not included in the general definition of a topological space. He writes (my translation from the Russian):

In subsequent development (after Hausdorff) of axioms of topological spaces, various notions were chosen as basic: closure (Kuratowski), open set (Aleksandrov), closed set (Sierpinski). All these approaches led to the same now universally accepted notion of a topological space ($T$-space), and undisputable credit for this belongs to Kuratowski.

Answer 2

What are the different ways that topologies have been presented or defined over time.

Topology is about a structure of cohesion on the points of a space. The question of what constitutes cohesion has been asked since earliest times, essentially arising from the paradoxes of Zeno. Aristotle thought of the physical continuum in terms of potentiality and actuality.

The same question was tackled by Hegel who thought of a moving point as being here and there at the same time. This is worth considering since an open set in pointless topology, is here and also over there. I mean by that, it is not localised to a point. The question of what constitutes the physical continuum was tackled in Weyls Space, Time & Matter, first published in 1918.

This sense of a 'motion' culminating in a 'limit' is axiomatised via nets and which generalises the definition of a limit of a sequence. This is exactly equivalent to the traditional notion of a topology by open or closed sets and deserves to be better known that it is now, especially given that limits are usually introduced at school and hence, pedagogically speaking, generalised limits - aka nets - are a natural way to introduce topology.

We have the classical notion of a topology in terms of nets (or filters), opens (or closeds) and interior (or closure operators). There are more general notion of topologies such as pseudo-topologies and convergence spaces. They aim to have better categorical properties. For example, Top, the category of topological spaces does not have exponentials, that is a natural topology on mapping spaces. Whereas Conv, the category of convergence spaces, being a quasi-topos, does; and moreover, Top embeds in it as a full subcategory. This means Conv extends the notion of topology.

But there is more to the question of cohesion than simpe topology. We can also ask how smoothly do they cohere. This of course is differentiability and this has also been axiomatised. For example, there is diffeology, which was put forward by Souriau in the 1980s, and which aims to do for smoothness what topology does for continuity. There is also the notion of bornology, which does the same for boundedness - in fact, one mathematician has said that in functional analysis, bornology was more natural than topology. In fact, there is also the notion of convexology , a neologism for a similar notion for convex structures. This might seem to be far from the notion of cohesion but it's possible that the fine structure of a space may be convex as in locally convex spaces and where the convexity is crucial to define a well-behaved notion of differentiability and hence of smoothness.

It's worth adding that categories can be topologised. Here, what is used is an abstraction of the notion of covering from topology and what is called a Grothendieck topology. This was formulated by Grothendieck in the early 1960s.

It's probably also worth saying that uniform continuity had a separate axiomatic development with the development of uniform spaces and uniformities. And also that completeness, which is about the fine structure of points; in particular, that there are no 'missing' points and hence 'complete' also has a separate axiomatix treatment as Cauchy spaces. The name here obviously derives from Cauchy sequences or nets which are traditionally used to define the completeness of a metric space, or more generally, of a metrisable topological space.

Answer 3

intuition behind a neighborhood system.

A version of this that I remember, leading up to Hausdorff (1914) defining his neighborhood axioms.

There was a notion (nowadays called a "manifold") where you specify a system of "charts" ... subsets of your space, and each of these subsets is homeomorphic to an open set in Euclidean space $\mathbb R^n$. Then you define "continuous" (as well as "dfferentiable" etc.) using the corresponding property in $\mathbb R^n$. But at some point Hausdorff noticed that to define "continuous" you could ignore the structure of $\mathbb R^n$: if you allow arbitrarily small "neighborhoods" then you can define continuity just in terms of whether certain neighborhoods of your domain map into certain other neighborhoods of your codomain, without referring to the $\mathbb R^n$ structure at all. Writing this abstractly led to the "Hausdorff space", entirely specified in terms of "neighborhoods".

Later, when other abstract notions of topology were investigated, it turned out in many cases that Hausdorff space was the same thing, if you add the one extra axiom, now known as the Hausdorff axiom.